A MEMOIR ON CUBIC SURFACES. 
419 
412] 
118. Take m x , m 2 as the roots of the equation (m — l) 2 = 4am, so that m x + m 2 = 2 + 4a, 
m x m 2 = 1, then the planes are 
X = 0, 
[ 7] 
k^ 
N 
II 
o 
[ 8] 
Z =0, 
[ 9] 
F + £ +X = 0, 
[ 12] 
F+ A r + F = 0, 
[ 34] 
F+Z + F = 0, 
[ 56] 
k^ 
II 
I 
i—* 
.N 
[ 13] 
F = (m 2 — 1) X, 
[ 24] 
F = (m 2 - 1) X 
[ 15] 
F = (mj — 1) Z, 
[ 25] 
Y = (m, - 1) F, 
[ 46] 
Y= (m 2 — 1) F, 
[ 35] 
kH 
N 
II 
o 
[789] 
F+X+Z+ F= 0, 
[789] 
119. And the lines are 
a 
b 
c 
/ 
9 
h 
equations may be written 
0 
0 
0 
0 
0 
1 
(1) 
o 
II 
k 
o 
II 
0 
0 
0 
1 
0 
0 
(8) 
Z =0, 7 = 0 
0 
1 
0 
0 
0 
0 
(9) 
o 
II 
k 
o' 
II 
K 
1 
1 
1 
0 
0 
0 
(7) 
Y + Z + X = 0, 17 = 0 
0 
0 
1 
-1 
1 
0 
(8) 
7+ A" + F = 0, Z =0 
1 
0 
0 
0 
- 1 
1 
(9) 
Y + Z + 17=0, AT=0 
0 
0 
0 
1 
1 
1 
(1) 
7= (m x — 1) X =(m 2 — l)Z 
m x — 1 
m 2 — 1 
0 
0 
0 
1 
1 
1 
(2) 
Y=(m 2 -l)X=(m x -l)Z 
m 2 - 1 
m x — 1 
-1 
1 
0 
0 
0 
1 
(3) 
7 = (m 2 — 1) 17 = (m x - 1) X 
m 1 — 1 
m 2 - 1 
-1 
1 
0 
0 
0 
1 
(4) 
7=(m 1 -l) W=(m 2 -l)X 
m 2 — 1 
m x — 1 
0 
1 
1 
1 
0 
0 
(5) 
Y = (m x — \)Z= (m 2 — 1) 17 
m x - 1 
m 2 — 1 
0 
1 
1 
1 
0 
0 
(6) 
7= (m 2 — 1) A - = (wq - 1) F 
m 2 — 1 
m x — 1